That Mr. Romney has not yet named his vice-presidential nominee has created an opening for social and economic conservatives to pressure him publicly, and they have taken the opportunity to make an aggressive case for Representative Paul D. Ryan of Wisconsin.

In rallying around Mr. Ryan, a champion of cutting government spending and reining in the costs of programs like Medicare and Medicaid, conservatives are calling for Mr. Romney to select someone who can push their fiscal agenda, but they also are setting the stage for a possible letdown on the right if Mr. Romney chooses someone else in his race against President Obama. A strongly worded Wall Street Journal editorial on Thursday urged Mr. Romney to pick Mr. Ryan, saying he “best exemplifies the nature and stakes of this election.”

The editorial follows a fresh wave of public pressure from other conservative outlets for Mr. Romney to erase doubts about his commitment to conservative causes — an issue that has dogged Mr. Romney since his days campaigning as a liberal Republican for the Senate in Massachusetts.

“The conservative base of the party is so concerned about Obama and his approach to government that they are going to vote for Romney,” said John Brabender, who was Rick Santorum’s chief strategist during his nominating fight with Mr. Romney. “The question is, are they going to make 10 phone calls to their friends and relatives because they care so passionately? That’s going to be somewhat of a challenge.”

The Weekly Standard on Thursday urged Mr. Romney to embrace the conservative principles in Mr. Ryan’s budget — and Mr. Ryan himself as his pick for vice president — predicting that Democrats will attack him for going after entitlement programs anyway.

“Romney, and Republicans, will be running on the Romney-Ryan plan no matter what,” The Weekly Standard wrote. That view was echoed by Newt Gingrich, who lost a bid for the Republican nomination to Mr. Romney.

“If Romney needs to defend the Paul Ryan budget, there’s no better way than to put Paul Ryan up front to defend it,” he said, adding that Mr. Ryan could help Mr. Romney in culturally conservative parts of the industrial Midwest.

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Tim Pawlenty is said to be on Mitt Romney’s shortlist for possible vice-presidential nominees.Credit
Max Whittaker for The New York Times

The not-so-subtle campaign among conservatives on Mr. Ryan’s behalf may be moot if Mr. Romney has already made up his mind about a running mate, as some political observers believe. It is possible that Mr. Romney could announce his pick as early as this weekend, while on a scheduled bus tour through swing states. Speaking with Chuck Todd of NBC News on Thursday, Mr. Romney said his choice must add “something to the political discourse about the direction of the country.”

The names on his shortlist are said to include Mr. Ryan; Senator Rob Portman of Ohio; and Tim Pawlenty, the former governor of Minnesota.

But the loud, public calls on Mr. Ryan’s behalf underscore the wariness with which conservatives have treated Mr. Romney. They suggest that some conservatives remain eager for Mr. Romney to demonstrate that he is, in fact, one of them.

Mr. Romney’s campaign added to the concern in the last couple of days on the issue of health care — a source of lingering suspicion among conservatives because Mr. Romney, as governor of Massachusetts, once championed an individual mandate very similar to the one in Mr. Obama’s health law.

In defending Mr. Romney against an attack ad highlighting a cancer patient whose husband lost his job at a steel mill owned by Mr. Romney’s firm, the campaign praised the Massachusetts health care plan. In doing so, the campaign came perilously close in the minds of some conservatives to sounding like the president.

“If people had been in Massachusetts, under Governor Romney’s health care plan, they would have had health care,” Andrea Saul, a spokeswoman for Mr. Romney, said Tuesday. That prompted Erick Erickson, of the conservative blog Redstate, to write on Twitter: “OMG. This might just be the moment Mitt Romney lost the election. Wow.”

But in Iowa earlier, Mr. Romney went out of his way to talk about his health care experience: “We’ve got to do some reforms in health care, and I have some experience doing that, as you know.”

Polls suggest that Mr. Romney’s transition from primary candidate to presumptive nominee has deeply unified Republicans around his candidacy. Conservative support for Mr. Romney is strong in part because of a dislike of President Obama.

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Rob Portman is said to be on Mitt Romney's shortlist for possible vice-presidential nominees. Mr. Romney could announce his choice as early as this weekend, while on a bus tour through swing states.Credit
Philip Scott Andrews/The New York Times

But Mr. Romney could energize conservatives and spur turnout even more by picking someone seen by the most ardent members of that group as someone who will be an uncompromising advocate for conservative principles inside a Romney White House.

The argument is about politics and about governing.

Advocates for Mr. Ryan argue that he would be a boon to Mr. Romney on the ballot by cementing in voters’ minds an economic vision for the country that is very different than Mr. Obama’s.

“The House budget chairman has defined those stakes well as a generational choice about the role of government and whether America will once again become a growth economy or sink into interest-group dominated decline,” The Journal wrote.

But conservatives are also looking past November to the kind of White House they want should Mr. Romney win.

For some of those conservatives, a Romney administration stocked with moderate Republicans is almost as bad as a second term for Mr. Obama. And for some of them, even Mr. Ryan is not conservative enough.

Richard Viguerie, the conservative direct mail pioneer, called him “a nice guy” but said, “He is not Tea Party.”

“He’s part of the Washington crowd,” Mr. Viguerie said Thursday. “His solution is basically a version of Washington, D.C., insanity. His proposal doesn’t balance the budget for 28 years.”

Mr. Viguerie said a ticket made up of Washington stalwarts would not motivate conservatives to work hard for Mr. Romney in the fall.

“Since Romney was not the first, second or third choice of most grass-roots conservatives and he spent massive amounts of money trashing conservative candidates, there is a lot of healing that needs to take place,” he said.

Michael D. Shear reported from Washington, and Trip Gabriel from New York.

A version of this article appears in print on August 10, 2012, on page A8 of the New York edition with the headline: A Conservative Bid for Ryan to Be Romney’s Running Mate. Order Reprints|Today's Paper|Subscribe